A&A 592, A156 (2016) Astronomy DOI: 10.1051/0004-6361/201628558 & c ESO 2016 Astrophysics

The Solar Twin Planet Search IV. The Sun as a typical rotator and evidence for a new rotational braking law for Sun-like stars?,??

Leonardo A. dos Santos1, 2, Jorge Meléndez1, José-Dias do Nascimento Jr.3, 4, Megan Bedell2, Iván Ramírez5, Jacob L. Bean2, Martin Asplund6, Lorenzo Spina1, Stefan Dreizler7, Alan Alves-Brito8, and Luca Casagrande6

1 Universidade de São Paulo, Departamento de Astronomia do IAG/USP, Rua do Matão 1226, Cidade Universitária, 05508-900 São Paulo, SP, Brazil e-mail: [email protected] 2 University of Chicago, Department of Astronomy and Astrophysics, IL 60637, USA 3 Universidade Federal do Rio Grande do Norte, 59072-970 Natal, RN, Brazil 4 Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA 5 University of Texas, McDonald Observatory and Department of Astronomy at Austin, USA 6 The Australian National University, Research School of Astronomy and Astrophysics, Cotter Road, Weston, ACT 2611, Australia 7 University of Göttingen, Institut für Astrophysik, Germany 8 Universidade Federal do Rio Grande do Sul, Instituto de Física, Av. Bento Gonçalves 9500, 90650-002 Porto Alegre, RS, Brazil

Received 19 March 2016 / Accepted 20 June 2016

ABSTRACT

Context. It is still unclear how common the Sun is when compared to other similar stars in regards to some of its physical properties, such as rotation. Considering that gyrochronology relations are widely used today to estimate ages of stars in the main sequence, and that the Sun is used to calibrate it, it is crucial to assess whether these procedures are acceptable. Aims. We analyze the rotational velocities, limited by the unknown rotation axis inclination angle, of an unprecedented large sample of solar twins to study the rotational evolution of Sun-like stars, and assess whether the Sun is a typical rotator. Methods. We used high-resolution (R = 115 000) spectra obtained with the HARPS spectrograph and the 3.6 m telescope at La Silla Observatory. The projected rotational velocities for 81 solar twins were estimated by line profile fitting with synthetic spectra. Macro- turbulence velocities were inferred from a prescription that accurately reflects their dependence with effective temperature and lumi- nosity of the stars. Results. Our sample of solar twins include some spectroscopic binaries with enhanced rotational velocities, and we do not find any nonspectroscopic binaries with unusually high rotation velocities. We verified that the Sun does not have a peculiar rotation, but the solar twins exhibit rotational velocities that depart from the Skumanich relation. Conclusions. The Sun is a regular rotator when compared to solar twins with a similar age. Additionally, we obtain a rotational braking law that better describes the stars in our sample (v ∝ t−0.6) in contrast to previous, often-used scalings. Key words. Sun: rotation – stars: solar-type – stars: rotation – stars: fundamental parameters

1. Introduction typical (i.e., an average Sun-like star)? If the Sun is common, it would mean that life does not require a special star for it to The Sun is the best-known star to astronomers and is commonly flourish, eliminating the need to evoke an anthropic reasoning to used as a template in the study of other similar objects. Yet, there explain it. are still some of its aspects that are not well understood and that are crucial for a better understanding of how stars and, conse- In an effort to assess how typical the Sun is, Robles et al. quently, how planetary systems and life evolve: how do the more (2008) compared 11 of its physical parameters with nearby complex physical parameters of a Sun-like star, such as rotation stars, and concluded that the Sun is, in general, typical. Al- and magnetic activity, change with time? Is the Sun unique or though they found it to be a slow-rotator against 276 F8 – K2 (within ±0.1 M ) nearby stars, this result may be rendered in- ? Based on observations collected at the European Organisation conclusive owing to unnacounted for noise that is caused by for Astronomical Research in the Southern Hemisphere under ESO different masses and ages in their sample. Other studies have programs 188.C-0265, 183.D-0729, 292.C-5004, 077.C-0364, 072.C- suggested that the Sun rotates either unusually slowly (Smith 0488, 092.C-0721, 093.C-0409, 183.C-0972, 192.C-0852, 091.C- 1979; Leão et al. 2015) or regularly for its age (Soderblom 0936, 089.C-0732, 091.C-0034, 076.C-0155, 185.D-0056, 074.C-0364, 075.C-0332, 089.C-0415, 60.A-9036, 075.C-0202, 192.C-0224, 090.C- 1983, 1985; Gray 1984; Gustafsson 1998; Barnes 2003), but 0421 and 088.C-0323. none of these investigations comprised stars that are very sim- ?? Full Table3 is only available at the CDS ilar to the Sun, therefore preventing a reliable comparison. In via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via fact, with Kepler and CoRoT, it is now possible to obtain http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/592/A156 precise measurements of rotation periods, masses and ages of

Article published by EDP Sciences A156, page 1 of8 A&A 592, A156 (2016) stars in a very homogeneous way (e.g., Ceillier et al. 2015; analysis that we perform (see Fig.1 for an illustration of the sub- do Nascimento et al. 2012; Chaplin et al. 2014), but they gener- tle effects of rotation in stellar spectra of Sun-like stars). ally lack high-precision stellar parameters, which are accessible through spectroscopy. The challenging nature of these observa- tions limited ground-based efforts to smaller, but key stellar sam- ples (e.g., Pizzolato et al. 2003; Strassmeier et al. 2012). 2. Working sample The rotational evolution of a star plays a crucial role in Our sample consists of bright solar twins in the Southern Hemi- stellar interior physics and habitability. Previous studies pro- sphere, which were mostly observed in our HARPS Large Pro- posed that rotation can produce extra mixing that is respon- gram (ID: 188.C-0265) at the European Southern Observatory sible for depleting the light elements Li and Be in their at- (ESO) that aimed to search for planetary systems around stars mospheres (Pinsonneault et al. 1989; Charbonnel et al. 1994; very similar to the Sun (Ramírez et al. 2014; Bedell et al. 2015; Tucci Maia et al. 2015), which could explain the disconnection Tucci Maia et al. 2016, Papers I, II and III, respectively, of the between meteoritic and solar abundances of Li (Baumann et al. series The Solar Twin Planet Search). These stars are loosely 2010). Moreover, rotation is highly correlated with mag- defined as those that have T eff, log g and [Fe/H] inside the in- netic activity (e.g., Noyes et al. 1984; Soderblom et al. 1993; tervals ±100 K, ±0.1 [cgs] and 0.1 dex, respectively, around Baliunas et al. 1995; Mamajek & Hillenbrand 2008), and this the solar values. It has been shown that these limits guaran- trend is key to understanding how planetary systems and life tee ∼0.01 dex precision in the relative abundances derived us- evolve in face of varying magnetic activity and energy outputs ing standard model atmosphere methods and that the system- by solar-like stars during the main sequence (Guinan & Engle atic uncertainties of that analysis are negligible within those 2009; Ribas et al. 2005; do Nascimento et al. 2016). ranges (Bedell et al. 2014; Biazzo et al. 2015; Saffe et al. 2015; A theoretical treatment of rotational evolution from first Yana Galarza et al. 2016). In total, we obtained high-precision principles is missing, so we often rely on empirical studies to spectra for 73 stars and used data from 9 more targets observed make inferences about it. One of the pioneer efforts in this in other programs; all of these overlapped the sample of 88 stars endeavor produced the well-known Skumanich relation v ∝ from Paper I. We used the spectrum of the Sun (reflected light t−1/2, where v is the rotational velocity and t is the stellar from the Vesta asteroid) from the ESO program 088.C-0323, age (Skumanich 1972), which describes the rotational evolu- which was obtained with the same instrument and configuration tion of solar-type stars in the main sequence, and can be de- as the solar twins. rived from the loss of angular momentum due to magnetized The ages of the solar twin sample span between 0−10 Gyr stellar winds (e.g., Kawaler 1988; Charbonneau 1992; Barnes and are presented in Table3. They were obtained by 2003; Gallet & Bouvier 2013). This relation sparked the devel- Tucci Maia et al.(2016) using Yonsei-Yale isochrones (Yi et al. opment of gyrochronology, which consists in estimating stel- 2001) and probability distribution functions as described in lar ages based on their rotation, and it was shown to provide Ramírez et al.(2013, 2014). Uncertainties are assumed to be a stellar clock as good as chromospheric ages (Barnes 2007). In symmetric. These ages are in excellent agreement with those Skumanich-like relations, however, the Sun generally falls on the obtained in Paper I, with a mean difference of −0.1 ± 0.2 Gyr curve (or plane, if we consider dependence on mass) defined by (see footnote 5 in Paper III). We adopted 4.56 Gyr for the so- the rotational braking law by design. Thus it is of utmost impor- lar age (Bahcall et al. 1995). The other stellar parameters (Teff, tance to assess how common the Sun is to correctly calibrate it. log g, [Fe/H] and microturbulence velocities vt) were obtained by Subsequent studies have proposed modifications to this Ramírez et al.(2014). The stellar parameters of HIP 68468 and paradigm of rotation and chromospheric activity evolution (e.g., HIP 108158 were updated by Tucci Maia et al.(2016). Soderblom et al. 1991; Pace & Pasquini 2004), exploring rota- Our targets were observed at the HARPS spectrograph tional braking laws of the form v ∝ t−b. The formalism by (Mayor et al. 2003), which is fed by ESO’s 3.6 m telescope at Kawaler(1988) shows that this index b can be related to the La Silla Observatory. When available publicly, we also included geometry of the stellar magnetic field, and that the Skumanich all observations from other programs in our analysis in order index (b = 1/2) corresponds to a geometry that is slightly more to increase the signal-to-noise ratio (S/N) of our spectra. How- complex than a simple radial field. It also dictates the depen- ever, we did not use observations for 18 Sco (HIP 79672) from dence of the angular momentum on the rotation rate and, in prac- May 2009, owing to their instrumental artifacts, and we did not tice, it determines how early the effects of braking are felt by a include observations post-HARPS upgrade (June 2015) when model. Such prescriptions for rotational evolution have a general combining the spectra – they had a different shape in the red agreement for young ages up to the solar age (see Sood et al. side, and since there were few observations, we chose not to use 2016; Amard et al. 2016, and references therein), but the evo- them to eliminate eventual problems with combination and nor- lution for older ages still poses an open question. In particular, malization. Our initial plan was to use the observations from the van Saders et al.(2016) suggested that stars undergo a weakened MIKE spectrograph, as described by the Paper I. However, we magnetic braking after they reach a critical value of the Rossby decided to use the HARPS spectra due to its higher spectral re- number, thus explaining the stagnation trend observed on the ro- solving power. tational periods of older Kepler stars. The wavelength coverage for the observations ranged from In order to assess how typical the Sun is in its rotation, our 3780 to 6910 Å, with a spectral resolving power of R = λ/∆λ = study aims to verify whether the Sun follows the rotational evo- 115 000. Data reduction was performed automatically with the lution of stars that are very similar to it, which is an objective HARPS Data Reduction Software (DRS). Each spectrum was that is achieved by precisely measuring their rotational velocities divided into two halves, corresponding to the mosaic of two de- and ages. We take advantage of an unprecedented large sample tectors (one optimized for blue and other for red wavelengths). of solar twins (Ramírez et al. 2014) using high signal to noise In this study we only worked with the red part (from 5330 to (S/N > 500) and high-resolution (R > 105) spectra, which pro- 6910 Å) because of its higher S/N and the presence of cleaner vides us with precise stellar parameters and is essential for the lines. The correction for radial velocities was performed with

A156, page 2 of8 L. A. dos Santos et al.: The Solar Twin Planet Search. IV.

1.0 1.0

0.9 0.9

0.8 λ d

λ 0.8 λ d F

λ 0.7 F

0.6 0.7

0.5 0.6 HIP 8507 0.4 HIP 19911

6021.2 6021.4 6021.6 6021.8 6022.0 6022.2 6022.4 6151.3 6151.4 6151.5 6151.6 6151.7 6151.8 6151.9 6152.0 λ (Å) λ (Å) Fig. 1. Comparison of the spectral line broadening between two solar Fig. 2. Example of line profile fitting for the Fe I feature at 6151.62 Å in twins with different projected rotational velocities. The wider line cor- the spectrum of the Sun. The continuous curve is the synthetic spectrum, responds to HIP 19911, with v sin i ≈ 4.1 km s−1, and the narrower line and the open circles are the observed data. comes from HIP 8507, with v sin i ≈ 0.8 km s−1.

Table 1. Line list used in the projected stellar rotation measurements. atmosphere models by Castelli & Kurucz(2004), with interpo- lations between models performed automatically by the Python 2 package qoyllur-quipu (see Ramírez et al. 2014). The instru- Wavelength Z Exc. pot. log (g f ) vmacro EW −1 mental broadening is taken into account by the spectral synthe- (Å) (eV) (km s ) (Å) sis. We used the stellar parameters from Tucci Maia et al.(2016) 6027.050 26 4.076 –1.09 3.0 0.064 and microturbulence velocities from Ramírez et al.(2014). 6151.618 26 2.176 –3.30 3.2 0.051 6165.360 26 4.143 –1.46 3.1 0.045 Macroturbulence velocities (vmacro) were calculated by scaling the solar values, line by line (see Sect. 3.1). An estimation of 6705.102 26 4.607 –0.98 3.6 0.047 3 6767.772 28 1.826 –2.17 2.9 0.079 the rotational velocities was performed with our own algorithm that makes automatic measurements for all spectral lines for each Notes. EW are the equivalent widths and vmacro are the macroturbulence star. We applied fine-tuning corrections by eye for the nonsat- velocities measured as in Sect. 3.1. isfactory automatic line profile fittings, and quote v sin i as the mean of the values measured for the five lines. See Sects. 3.1 and 3.2 for a detailed description on rotational velocities esti- the task dopcor from IRAF1, using the values obtained from the mation and their uncertainties. Figure2 shows an example of cross-correlation function (CCF) of the pipeline. The different spectral line fitting for one feature in the Sun. observations were combined with IRAF’s scombine. The result- ing average (of the sample) signal to noise ratio was 500 around 6070 Å. The red regions of the spectra were normalized with 3.1. Macroturbulence velocities ∼30th order polynomial fits to the upper envelopes of the entire v red range, using the task continuum on IRAF. We made sure We tested the possibility of measuring macro (radial-tangential v i that the continuum of the stars were consistent with the Sun’s. profile) simultaneously with sin , but even when using the ex- Additionally, we verified that errors in the continuum determi- tremely high-resolution spectra of HARPS, it is difficult to dis- nation introduce uncertainties in v sin i lower than 0.1 km s−1. entangle these two spectral line broadening processes; this is probably because of the low values of these velocities. Macro- turbulence has a stronger effect on the wings of the spectral 3. Methods lines, but our selection of clean lines still has some contamina- tion that requires this high-precision work to be carried out by We analyzed five spectral lines, four due to Fe I and one to Ni I eye. Some stars show more contamination than others, compli- (see Table1; equivalent widths were measured using the task cating the disentaglement. Fortunately, the variation of macro- splot in IRAF), which were selected for having low levels of turbulence with effective temperature and luminosity is smooth contamination by blending lines. The rotational velocity of a star (Gray 2005), so that precise values of vmacro could be obtained can be measured by estimating the spectral line broadening that by a calibration. Thus we adopted a relation that fixes macrotur- is due to rotation. The rotation axes of the stars are randomly bulence velocities to measure v sin i with high precision using an oriented, thus the spectroscopic measurements of rotational ve- automatic code, which provides the additional benefits of repro- locity are a function of the inclination angle (v sin i). ducibility and lower subjectivity. We estimate v sin i for our sample of solar twins using the The macroturbulence velocity is known to vary for different 2014 version of MOOG Synth (Sneden 1973), adopting stellar spectral lines (Gray 2005), so for our high-precision analysis we do not adopt a single value for each star. Instead, we measure 1 IRAF is distributed by the National Optical Astronomy Observato- the vmacro for the Sun in each of the spectral lines from Table1 ries, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Sci- 2 Available at https://github.com/astroChasqui/q2 ence Foundation. 3 Available at https://github.com/RogueAstro/PoWeRS

A156, page 3 of8 A&A 592, A156 (2016)

Table 2. Simultaneous measurements of rotational and macroturbulence and use these values to scale the vmacro for all stars in our sample using the following equation4: velocities of stars in the extremes of log g from our sample of solar twins. ∗ −7 2 vmacro,λ = vmacro,λ − 0.00707 Teff + 9.2422 × 10 Teff
+ 10.0 + k1 log g − 4.44 + k2 Star v sin i vmacro Teff log g −1 −1 ≡ f (T ) + k log g − 4.44
+ k (1) (km s ) (km s ) eff 1 2 HIP 115577 0.95 ± 0.05 3.35 ± 0.09 5699 4.25 ± ± where vmacro,λ is the macroturbulence velocity of the Sun for a HIP 65708 1.20 0.09 3.55 0.08 5755 4.25 HIP 74432 1.40 ± 0.03 3.35 ± 0.08 5684 4.25 given spectral line, Teff and log g are the effective temperature HIP 118115 1.40 ± 0.10 3.43 ± 0.09 5808 4.28 and gravity of a given star, respectively, k1 is a proportionality factor for log g and k is a small correction constant. HIP 68468 1.75 ± 0.07 3.70 ± 0.08 5857 4.32 2 HIP 41317 1.55 ± 0.03 3.10 ± 0.06 5700 4.38 This formula is partly based on the relation derived by Sun 1.75 ± 0.07 3.30 ± 0.06 5777 4.44 Meléndez et al.(2012; Eq. (E.1) in their paper) from the trend of HIP 105184 2.50 ± 0.03 3.21 ± 0.08 5833 4.50 macroturbulence with effective temperature in solar-type stars HIP 10175 1.55 ± 0.06 3.05 ± 0.08 5738 4.51 described by Gray(2005). The log g-dependent term (a proxy HIP 114615 2.20 ± 0.03 3.25 ± 0.08 5816 4.52 for luminosity) comes from the empirical relation derived by HIP 3203 3.90 ± 0.03 3.40 ± 0.10 5850 4.52 Doyle et al.(2014; Eq. (8) in their paper), and is based on spec- troscopic measurements of vmacro of Kepler stars, which were disentangled from v sin i using asteroseismic estimates of the projected rotational velocities. Doyle et al. obtained a value 0.6 for the proportionality factor k1 of −2.0. Their uncertainties on v , however, were on the order of 1.0 km s−1. Thus, we de- This work macro Doyle et al. (2014) 0.4 cided to derive our own values of k1 and k2 by simultaneously ) 1 measuring vmacro and v sin i of a subsample of solar twins. − s This subsample was chosen to contain only single stars or vi- 0.2 m k sual binaries mostly in the extremes of log g (4.25−4.52) in our (

) f f entire sample. We assume these values have a linear relation- e 0.0 T ( ship with vmacro inside this short interval of log g. We used as a f − o first guess the values of v sin i and vmacro from a previous, cruder r 0.2 c a estimation we made, and performed line profile fits by eye us- m v ing MOOG Synth. The velocities in Table2 are the median of 0.4 the values measured for each line and their standard error. These v sin i are not consistently measured in the same way as the fi- 0.6 nal results. The rotational velocity broadening was calculated by our own code (see Sect. 3.2 for details). By performing a linear 0.25 0.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 logg 4. 44 (cgs) fit in the vmacro − f (Teff) versus log g − 4.44 relation ( f com- − prises all the Teff-dependent terms, the macroturbulence velocity Fig. 3. Linear relation between vmacro and log g (a proxy for luminosity) of the Sun and the known constant on Eq. (1)), we obtain that for the stars on Table2. See the definition of f (Teff ) in Sect. 3.1. The k1 = −1.81±0.26 and k2 = −0.05±0.03 (see Fig.3). For the stars orange continuous line represents our determination of a proportional- −1 farthest from the Sun in log g from our sample, these values of k1 ity coefficient of −1.81 and a vertical shift of −0.05 km s . The black −1 and k2 would amount to differences of up to ±0.4 km s in their dashed line is the coefficient found by Doyle et al.(2014). The light macroturbulence velocities, therefore it is essential to consider gray region is a composition of 200 curves with parameters drawn from a multivariate Gaussian distribution. The Sun is located at the origin. the luminosity effect on vmacro for accurate v sin i determinations. To obtain the macroturbulence velocities for the Sun to use in Eq. (1), we forced the rotational velocity of the Sun to 1.9 km s−1

(Howard & Harvey 1970), and then estimated values of vmacro,λ by fitting each line profile using MOOG Synth, and the results star, the code automatically corrects the spectral line shift and are shown in Table1. We estimated the error in determining the continuum. The first is performed by fitting a second-order −1 vmacro,λ to be ±0.1 km s . Since Eq. (1) is an additive scaling, polynomial to the kernel of a line and estimating the distance of the error for vmacro of all stars is the same as in the Sun. The un- the observed line center from the laboratory value. Usually, the certainties in stellar parameters have contributions that are neg- spectral line shift corrections were on the order of 10−2 Å, cor- −1 ligible compared to the ones introduced by the error in vmacro. responding to 0.5 km s in the wavelength range we worked on. This is a reasonable shift that likely arises from a combination of granulation and gravitational redshift effects, which are of simi- 3.2. Rotational velocities lar magnitude. The continuum correction for each line is defined Our code takes as input the list of stars and their parameters (ef- as the value of a multiplicative factor that sets the highest flux fective temperature, surface gravity, metallicity and microturbu- inside a radius of 2.5 Å around the line center to 1.0. The multi- lence velocities obtained on Paper I), their spectra and the spec- plicative factor usually has a value inside the range 1.000±0.002. tral line list in MOOG-readable format. For each line in a given The code starts with a range of v sin i and abundances and op- 4 In the future, it should be possible to calibrate macroturbulence timizes these two parameters through a series of iterations that velocities using 3D hydrodynamical stellar atmosphere models (e.g., measure the least-squares difference between the observed line Magic et al. 2013) by using predicted 3D line profiles (without rota- and the synthetic line (generated with MOOG synth). Conver- tional broadening) as observations and determine which value of vmacro gence is achieved when the difference between the best and pre- is needed to reproduce them with 1D model atmospheres. vious solutions, for both v sin i and abundance, is less than 1%.

A156, page 4 of8 L. A. dos Santos et al.: The Solar Twin Planet Search. IV.

Table 3. Ages, the measured v sin i, and stellar parameters of the solar twins and the Sun.

Star Age σ v sin i σ [Fe/H] σ Teff σ log g σ vt σ vmacro Note (Gyr) (km s−1) (dex) (K) (cgs) (km s−1) (km s−1) Sun 4.56 . . . 2.04 0.12 0.000 . . . 5777 . . . 4.44 . . . 1.00 . . . 3.20 SS HIP 1954 4.87 0.97 1.79 0.13 –0.068 0.006 5717 5 4.46 0.02 0.96 0.02 2.90 ... HIP 3203 0.99 0.66 3.82 0.11 –0.087 0.008 5850 10 4.52 0.02 1.16 0.02 3.27 SS HIP 4909 1.23 0.77 4.01 0.11 +0.028 0.008 5854 10 4.50 0.02 1.12 0.02 3.33 SS HIP 5301 6.49 0.67 2.00 0.12 –0.064 0.004 5728 5 4.42 0.02 0.97 0.01 3.01 SS HIP 6407 1.49 0.66 2.30 0.13 –0.068 0.007 5764 8 4.52 0.01 0.97 0.02 2.96 SB I HIP 7585 3.29 0.51 1.90 0.15 +0.095 0.005 5831 5 4.43 0.01 1.02 0.01 3.37 ... HIP 8507 3.63 0.94 0.77 0.15 –0.096 0.006 5725 6 4.49 0.02 0.99 0.02 2.88 ... HIP 9349 1.43 0.76 2.25 0.11 +0.009 0.007 5810 8 4.50 0.02 1.07 0.02 3.16 ... HIP 10175 1.82 0.65 1.83 0.11 –0.007 0.005 5738 7 4.51 0.01 0.96 0.01 2.89 ... HIP 10303 5.48 0.56 0.77 0.16 +0.106 0.004 5725 4 4.40 0.01 0.98 0.01 3.04 ...

Notes. SS are stars belonging to the selected sample; SB I are single-lined spectroscopic binaries. This table is available in its entirety in machine- readable format at the CDS. A portion is shown here for guidance regarding its form and content.

Additionally, the code also forces at least ten iterations to avoid 4.5 falling into local minima. One of the main limitations of MOOG Synth for our analysis 4.0 is that it has a “quantized” behavior for v sin i: the changes in the 3.5 synthetic spectra occur most strongly in steps of 0.5 km s−1. This HIP 103983

behavior is not observed in varying the macroturbulence veloc- ) 3.0 1 − ities. Therefore, we had to incorporate a rotational broadening s 2.5 routine in our code that was separated from MOOG. We used m k (

the Eq. (18.14) from Gray(2005), in velocity space, to compute i 2.0 n 5 i the rotational profile s v 1.5 1/2 h 2i 1 h 2i 2(1 − ) 1 − (v/vL) + π 1 − (v/vL) 1.0 G(v) = 2 , (2) πvL(1 − /3) 0.5 0.0 where vL is the projected rotational velocity and  is the limb 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 1 darkening coefficient (for which we adopt the value 0.6). The ro- CCF v sini (km s− ) tational profile G(v) is then convolved with the MOOG synthetic profiles, which were generated with v sin i = 0. Fig. 4. Comparison between our estimated values of v sin i (y-axis) and those inferred from the cross-correlation funcion FWHM (x-axis). The The total uncertainties in rotational velocities are obtained spread around the 1:1 relation (black line) is σ = 0.20 km s−1. from the quadratic sum of the standard error of the five measure- ments and an uncertainty of 0.1 km s−1 introduced by the error in macroturbulence velocities. Systematic errors in the calcula- Pace & Pasquini(2004), Hekker & Meléndez(2007), tion of vmacro,λ for the stars do not significantly contribute to the v sin i uncertainties. The rotational velocities we measured and which resulted in the following calibration: v sin i = q h i 2 2 2 −1 their uncertainties are reproduced in Table3. (0.73 ± 0.02) FWHM − vmacro − (5.97 ± 0.01) km s (esti- Some of the stars in the sample show very low rotational mation performed with the MCMC code emcee6 velocities, most probably owing to the effect of projection Foreman-Mackey et al. 2013). The scatter between the mea- (see left panel of Fig.5). The achieved precision is validated sured v sin i and those estimated from CCF is σ = 0.20 km s−1 by comparison with the values of the full width at half max- (excluding the outlier HIP 103983). The typical uncertainty imum (FWHM) measured by the cross-correlation function in the rotational velocities we obtain with our method, that (CCF) from the data reduction pipeline, with the effects of is, line profile fitting with extreme high-resolution spectra, macroturbulence subtracted (see Fig.4). The spectroscopic is 0.12 km s−1, which implies that the average error of the binary star HIP 103983 has an unusually high v sin i when CCF FWHM v sin i scaling is 0.16 km s−1. This average error compared to the CCF FWHM, and a verification of its spectral could be significantly higher if the broadening by v is not line profiles reveals the presence of distortions, which are the macro accounted for. most probably caused by mismeasurement of rotational velocity (contamination of the combined spectrum by a companion; observations range from October 2011 to August 2012). 4. Binary stars We obtained a curve fit for the v sin i versus CFF FWHM (km s−1) using a similar relation as used by Melo et al.(2001), We identified 16 spectroscopic binaries (SB) in our sample of 81 solar twins by analyzing their radial velocities; some of 5 This is the same recipe adopted by the radiative transfer code MOOG. 6 Available at http://dan.iel.fm/emcee/current/

A156, page 5 of8 A&A 592, A156 (2016)

4.5 4.5 HIP 19911 4.0 4.0 HIP 29525 3.5 3.5

3.0 3.0 ) HIP 67620 ) 1 1

− HIP 43297 − s s 2.5 2.5

m HIP 73241 m k k ( (

i 2.0 i 2.0 n n i i s s

v 1.5 v 1.5

1.0 1.0

0.5 0.5

0.0 0.0 0 2 4 6 8 10 2 4 6 8 10 Age (Gyr) Age (Gyr) Fig. 5. Projected rotational velocity of solar twins as a function of their age. The Sun is represented by the symbol . Left panel: all stars of our sample; the orange triangles are spectroscopic binaries, blue circles are the selected sample and the blue dots are the remaining nonspectroscopic binaries. Right panel: the rotational braking law; the purple continuous curve is our relation inferred from fitting the selected sample (blue circles) −b of solar twins with the form v sin i = vf + m t , where t is the stellar age, and the fit parameters are vf = 1.224 ± 0.447, m = 1.932 ± 0.431, and b = 0.622 ± 0.354, with vf and b highly and positively correlated. The light gray region is composed of 300 curves that are created with parameters drawn from a multivariate Gaussian distribution defined by the mean values of the fit parameters and their covariance matrix. Skumanich’s law −1 (red × symbols, calibrated for vrot = 1.9 km s ) and the rotational braking curves proposed by do Nascimento et al.(2014, black dashed curve, smoothed) and Pace & Pasquini(2004, black dot-dashed curve) are plotted for comparison. these stars are reported as binaries by Tokovinin(2014a,b), than expected for single stars, we conclude that stellar multiplic- Mason et al.(2001), Baron et al.(2015). We did not find previ- ity is an important enhancer of rotation in Sun-like stars. Blue ous reports of multiplicity for the stars HIP 30037, HIP 62039 stragglers are expected to have a strong enhancement on rota- and HIP 64673 in the literature. Our analysis of variation in the tion owing to injection of angular momentum from the donor HARPS radial velocities suggest that the first two are probable companion. SBs, while the latter is a candidate. No binary shows a double- lined spectrum, but HIP 103983 has distortions that could be from contamination by a companion. The star HIP 64150 is a 5. The rotational braking law Sirius-like system with a directly observed white dwarf compan- ion (Crepp et al. 2013; Matthews et al. 2014). The sample from We removed from this analysis all the spectroscopic binaries Paper I contains another SB, HIP 109110, for which we could in order to correctly constrain the rotational braking. The non- not reliably determine the v sin i because of strong contamina- SB HIP 29525 displays a v sin i that is much higher than ex- −1 tion in the spectra, which is possibly caused by a relatively bright pected (3.85 ± 0.13 km s ), but it is likely that this is the companion. Thus, we did not include this star in our sample. result of an overestimated isochronal age (2.83 ± 1.06 Gyr). We decided to not include HIP 29525 in the rotational brak- Of these 16 spectroscopic binaries, at least four of ing determination because it is a clear outlier in our results. them (HIP 19911, 43297, 67620 and 73241) show unusually Maldonado et al.(2010) found X-ray and chromospheric ages of high v sin i (see the left panel of Fig.5). These stars also 0.55 and 0.17 Gyr, respectively, for HIP 29525. We then divided present other anormalities, such as their [Y/Mg] abundances the remaining 65 stars and the Sun in bins of 2 Gyr, and removed (Tucci Maia et al. 2016) and magnetic activity (Ramírez et al. all the stars that were below the 70th percentile of v sin i in each 2014; Freitas et al., in prep.). The solar twin blue straggler bin from this sample. Such a procedure can be justified because HIP 10725 (Schirbel et al. 2015), which is not included in our we verified that 30% of the stars should have sin i above 0.9 by sample, also shows a high v sin i for its age. We find that five doing a simple simulation with angles i drawn from a flat distri- of the binaries have rotational velocities below the expected for bution between 0 and π/2. This allowed us to select the stars that Sun-like stars, but this is most likely an effect of projection of the had the highest chance of having sin i above 0.9. In total, 21 so- rotational axes of the stars. For the remaining binaries, which lar twins and the Sun compose what we hereafter reference as follow the rotational braking law, it is again difficult to disen- the selected sample. Albeit this subsample is smaller, it has the tangle this behavior from the sin i, and a statistical analysis is advantage of mostly removing uncertainties on the inclination precluded by the low numbers involved. Tidal interactions be- angle of the stellar rotation axes7. We stress that the only reason tween companions that could potentially enhance rotation de- we can select the most probable edge-on rotating stars (i = π/2) pend on binary separation, which is unknown for most of these is because we have a large sample of solar twins in the first place. stars. They should be regular rotators, since they do not show We then proceeded to fit a general curve to the selected anormalities in chromospheric activity (Freitas et al., in prep.) or sample (see Fig.5) using the method of orthogonal distance [Y/Mg] abundances (Tucci Maia et al. 2016). Based on the information that at least 25% of the spectro- 7 This procedure can also allow for unusually fast-rotating stars (al- scopic binaries in our sample show higher rotational velocities though rare) with sin i below 0.9 to leak into the subsample.

A156, page 6 of8 L. A. dos Santos et al.: The Solar Twin Planet Search. IV. regression (ODR, Boggs & Rogers 1990), which takes the un- The rotational braking law we obtain produces a similar out- certainties on both v sin i and ages into account. This curve is come to that achieved by van Saders et al.(2016) for stars older −b a power law plus constant of the form v = vf + m t (the than the Sun, that is, a weaker rotational braking law after solar same chromospheric activity and v sin i vs. age relation used age than previously suggested. Our data also requires a differ- by Pace & Pasquini 2004; Guinan & Engle 2009), with v (rota- ent power-law index than the Skumanich index for stars younger −1 tional velocity) and vf (asymptotic velocity) in km s and t (age) than the Sun, accounting for an earlier decay of rotational veloc- in Gyr. ities up to 2 Gyr. We find that the best-fit parameters are vf = 1.224 ± 0.447, The main-sequence spin-down model by Kawaler(1988) m = 1.932 ± 0.431, and b = 0.622 ± 0.354 (see right panel of states that, for constant moment of inertia and radius during the Fig.5). These large uncertainties are likely due to i) the strong main sequence, we would have correlation between vf and b; and ii) the relatively limited num- −3/(4an) ber of data points between 1 and 4 Gyr, where the parame- veq ∝ t , (3) ters are most effective in changing the values of v. This limi- where veq is the rotational velocity at the equator and a and n tation is also present in past studies (e.g., van Saders et al. 2016; are parameters that measure the dependence on rotation rate and Barnes 2003; Pace & Pasquini 2004; Mamajek & Hillenbrand radius, respectively (see Eqs. (7), (8) and (12) in their paper). 2008; García et al. 2014; Amard et al. 2016). On the other hand, If we assume a dipole geometry for the stellar magnetic field our sample is the largest comprising solar twins and, therefore, 3 (Br ∝ B0r ), then n = 3/7. Furthermore, assuming that a = 1, should produce more reliable results. With more data points, we −7/4 −1.75 then Eq. (3) results in veq ∝ t = t . The Skumanich law could be able to use 1 Gyr bins instead of 2 Gyr in order to select −0.5 (veq ∝ t ) is recovered for n = 3/2, which is close to the case the fastest rotating stars, which would result in a better subsam- −0.38 ple for constraining the rotational evolution for young stars. of a purely radial field (n = 2, veq ∝ t ). A more extensive The relation we obtain is in contrast with some previous exploration of the configuration and evolution of magnetic fields studies on modeling the rotational braking (Barnes 2001, 2003; of solar twins is outside the scope of this paper, but our results Lanzafame & Spada 2015) which either found or assumed that suggest that the rotational rotational braking we observe on this the Skumanich law explains well the rotational braking of Sun- sample of solar twins stems from a magnetic field with an inter- like stars. The conclusions by van Saders et al.(2016) limit the mediate geometry between dipole and purely radial. range of validation up to approximately the solar age (4 Gyr) for stars with solar mass. When we enforce the Skumanich power- 6. Conclusions law index b = 1/2, we obtain a worse fit between the ages 2 and 4 Gyr and, not surprisingly, after the solar age as well. We analyzed the rotational velocities of 81 bright solar twins Our data and the rotational braking law that results from in the Southern Hemisphere and the Sun using extremely high- them show that the Sun is a normal star regarding its rota- resolution spectra. Radial velocities revealed that our sample tional velocity when compared to solar twins. However, they do contained 16 spectroscopic binaries, 3 of which (HIP 30037, not agree with the regular Skumanich law (Barnes 2007, red × 62039, 64673) were not listed as such in the literature. At least 5 symbols in Fig.5). We find a better agreement with the model of these stars show an enhancement on their measured v sin i, proposed by do Nascimento et al.(2014, black dashed curve in which is probably caused by interaction with their close-by com- Fig.5), especially for stars older than 2 Gyr. This model is thor- panions. They also present other anomalies in chemical abun- oughly described in Appendix A of do Nascimento et al.(2012). dances and chromospheric activities. We did not clearly identify In summary, this model uses an updated treatment of the insta- nonspectroscopic binary stars with unusually high rotational ve- bilities that are relevant to the transport of angular momentum, locities for their age. according to Zahn(1992) and Talon & Zahn(1997), with an ini- We selected a subsample of stars with higher chances of hav- 50 2 −1 tial angular momentum for the Sun J0 = 1.63 × 10 g cm s . ing their rotational axis inclination close to π/2 (almost edge-on) The rotational braking curve that corresponds to the model by in order to better constrain the rotational evolution of the solar do Nascimento et al.(2014) is computed using the output radii twins. We opted to use carefully measured isochronal ages for of the model, which vary from ∼1 R at the current solar age these stars because it is the most reliable method available for to 1.57 R at the age of 11 Gyr, and it changes significantly this sample. We finally conclude that the Sun seems to be a com- if we use a constant radius R = 1 R . This results in a more mon rotator, within our uncertainties, when compared to solar Skumanich-like rotational braking. twins, therefore it can be used to calibrate stellar models. Our result agrees with the chromospheric activity versus Moreover, we have found that the Skumanich law does not age behavior for solar twins obtained by Ramírez et al.(2014), describe well the rotation evolution for solar twins observed in 0 our data, which is a discrepancy that is stronger after the solar in which a steep decay of the RHK index during the first 4 Gyr was deduced (see Fig. 11 in their paper). The study by age. Therefore, we propose a new rotational braking law that Pace & Pasquini(2004) also suggests a steeper power-law index supports the weakened braking after the age of the Sun, and comes with a earlier decay in rotational velocities up to 2 Gyr (b = 1.47) than Skumanich’s (bS = 1/2) in the rotational brak- ing law derived from young open clusters, the Sun and M 67. than the classical Skumanich’s law. Interestingly, it also reveals As seen in Fig.5, however, their relation significantly overesti- an evolution that is more similar to the magnetic activity evolu- mates the rotational velocities of stars, especially for those older tion observed in Sun-like stars, which sees a steep decay in the than 2 Gyr. This is most probably caused by other line broad- first 3 Gyr and flattens near the solar age. Additionally, we sug- ening processes, mainly the macroturbulence, which were not gest that more high-precision spectroscopic observations of solar considered in that study. As we saw in Sect. 3.1, those processes twins younger and much older than the Sun could help us better introduce important effects that are sometimes larger than the constrain the rotational evolution of solar-like stars. rotational broadening. Moreover, a CCF-only analysis tends to Acknowledgements. L.d.S. thanks CAPES and FAPESP, grants No. 2014/26908- produce more spread in the v sin i than the more detailed analy- 1 and 2016/01684-9 for support. J.M. thanks FAPESP (2012/24392-2) for sup- sis we used. port. L.S. acknowledges support by FAPESP (2014/15706-9). We also would

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